[petsc-users] Implementing discontinuous Galerkin FEM?

[Andrew Ho]

I am trying to implement a discontinuous Galerkin discretization using the
PETSc DM features to handle most of the topology/geometry specific
functions. However, I'm not really sure which direction to approach this
from since DG is kind of a middle ground between finite volume and
traditional continuous Galerkin finite element methods.

It appears to me that if I want to implement a nodal DG method, then it
would be more practical to extend the PetscFE interface, but for a modal DG
method perhaps the PetscFV interface is better?

There are still a few questions that I don't know the answers to, though.

Questions about implementing nodal DG:

1. Does PetscFE support sub/super parametric element types? If so, how do I
express the internal node structure for a nodal DG method (say, for example
located at the abscissa of a Gauss-Lobatto quadrature scheme)?
2. How would I go about making the dataset stored discontinuous between
neighboring elements (specifically at shared nodes for a nodal DG method)?
3. Similar to 2, how would I handle boundary conditions? Specifically, I
need a layer of data space of just the boundary nodes (not a complete
"ghost" element), and these are the actual constrained points.

Questions about implementing modal DG:

A. What does specifying the quadrature object for a PetscFV object actually
do? Is it purely a surface flux integration quadrature? How does the
quadrature object handle simplex-type elements in 2D/3D?
B. How would I go about modifying the limiters to take into account these
multiple modes?

-- 
Andrew Ho
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